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On the Euler-Lagrange equation for a variational problem
On the convergence of solutions of the Leray-$\alpha $ model to the trajectory attractor of the 3D Navier-Stokes system
| 1. | Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetniy 19, Moscow 127994, GSP-4, Russian Federation, Russian Federation |
| 2. | Department of Mathematics and Department of Mechanics and Aerospace Engineering, University of California, Irvine, CA 92697, United States |
| [1] |
Aseel Farhat, M. S Jolly, Evelyn Lunasin. Bounds on energy and enstrophy for the 3D Navier-Stokes-$\alpha$ and Leray-$\alpha$ models. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2127-2140. doi: 10.3934/cpaa.2014.13.2127 |
| [2] |
Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113 |
| [3] |
Kuanysh A. Bekmaganbetov, Gregory A. Chechkin, Vladimir V. Chepyzhov, Andrey Yu. Goritsky. Homogenization of trajectory attractors of 3D Navier-Stokes system with randomly oscillating force. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2375-2393. doi: 10.3934/dcds.2017103 |
| [4] |
Milan Pokorný, Piotr B. Mucha. 3D steady compressible Navier--Stokes equations. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 151-163. doi: 10.3934/dcdss.2008.1.151 |
| [5] |
Luca Bisconti, Davide Catania. Remarks on global attractors for the 3D Navier--Stokes equations with horizontal filtering. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 59-75. doi: 10.3934/dcdsb.2015.20.59 |
| [6] |
Anne Bronzi, Ricardo Rosa. On the convergence of statistical solutions of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ vanishes. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 19-49. doi: 10.3934/dcds.2014.34.19 |
| [7] |
Tomás Caraballo, Antonio M. Márquez-Durán, José Real. Pullback and forward attractors for a 3D LANS$-\alpha$ model with delay. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 559-578. doi: 10.3934/dcds.2006.15.559 |
| [8] |
T. Tachim Medjo. Averaging of a 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external forces. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1281-1305. doi: 10.3934/cpaa.2011.10.1281 |
| [9] |
T. Tachim Medjo. A non-autonomous 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external force and its global attractor. Communications on Pure & Applied Analysis, 2011, 10 (2) : 415-433. doi: 10.3934/cpaa.2011.10.415 |
| [10] |
Gabriel Deugoue. Approximation of the trajectory attractor of the 3D MHD System. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2119-2144. doi: 10.3934/cpaa.2013.12.2119 |
| [11] |
Cung The Anh, Dang Thi Phuong Thanh, Nguyen Duong Toan. Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces. Evolution Equations & Control Theory, 2021, 10 (1) : 1-23. doi: 10.3934/eect.2020039 |
| [12] |
Gaocheng Yue, Chengkui Zhong. Attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 985-1002. doi: 10.3934/dcdsb.2011.16.985 |
| [13] |
Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141 |
| [14] |
Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241 |
| [15] |
Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1299-1306. doi: 10.3934/cpaa.2013.12.1299 |
| [16] |
Hui Chen, Daoyuan Fang, Ting Zhang. Regularity of 3D axisymmetric Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1923-1939. doi: 10.3934/dcds.2017081 |
| [17] |
A. V. Fursikov. Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 289-314. doi: 10.3934/dcds.2004.10.289 |
| [18] |
Francis Hounkpe, Gregory Seregin. An approximation of forward self-similar solutions to the 3D Navier-Stokes system. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4823-4846. doi: 10.3934/dcds.2021059 |
| [19] |
Chao Deng, Xiaohua Yao. Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 437-459. doi: 10.3934/dcds.2014.34.437 |
| [20] |
Ciprian G. Gal, T. Tachim Medjo. Approximation of the trajectory attractor for a 3D model of incompressible two-phase-flows. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2229-2252. doi: 10.3934/cpaa.2014.13.2229 |
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